Showing that the variation of an affine connection is a tensor

Question

How can i show that $\delta\Gamma_{\mu\nu}^{\rho}$ transforms like a tensor?

Metric compatibility is not assumed here.

That means

1) First i need to compute $\delta\Gamma$ first. To do that i need an explicit formula for it. But without metric compatibility, I don't which should i use. Like i get one from principle of equivalence :

$$ \Gamma_{\mu\nu}^{\lambda}=\frac{\partial^2\zeta^\alpha}{\partial x^\mu \partial x^\nu}\frac{\partial x^\lambda}{\partial\zeta^\alpha} $$

Where $\zeta$'s are local inertial coordinates obeying

$$ d\tau^2 = -\eta_{\alpha\beta}\zeta^\alpha\zeta^\beta $$

I'm confused about how to compute $\delta\Gamma$ from this.

2) After computing $\delta\Gamma$ I have to show it transforms like a (1,2) type tensor. Which i think i can manage if i can manage the 1.

Answer 1

Here we will use the physicist's definition of a tensor in terms of local coordinate transformations. Needless to say that this can also be cast in a mathematically manifestly covariant language.

It is well-known that the Christoffel symbol$^1$ $\Gamma^{\lambda}_{\mu\nu}$ does not transform as a tensor under a local coordinate transformation $x^{\mu} \to y^{\rho}=y^{\rho}(x)$, but rather with an inhomogeneous term, which is built from the second derivative of the coordinate transformation,

$$\tag{1}\frac{\partial y^{\tau}}{\partial x^{\lambda}} \Gamma^{(x)\lambda}_{\mu\nu} ~=~\frac{\partial y^{\rho}}{\partial x^{\mu}}\, \frac{\partial y^{\sigma}}{\partial x^{\nu}}\, \Gamma^{(y)\tau}_{\rho\sigma}+ \frac{\partial^2 y^{\tau}}{\partial x^{\mu} \partial x^{\nu}}. $$

Thus if we consider the (not necessarily infinitesimal) difference

$$\tag{2} \Delta \Gamma^{\lambda}_{\mu\nu}~:=~\Gamma^{(1)\lambda}_{\mu\nu}-\Gamma^{(2)\lambda}_{\mu\nu} $$

between two sets of Christoffel symbols, corresponding to two tangent-space connections $\nabla^{(1)}$ and $\nabla^{(2)}$, the difference will evidently transform as a (1,2) tensor

$$\tag{3}\frac{\partial y^{\tau}}{\partial x^{\lambda}} \Delta\Gamma^{(x)\lambda}_{\mu\nu} ~=~\frac{\partial y^{\rho}}{\partial x^{\mu}}\, \frac{\partial y^{\sigma}}{\partial x^{\nu}}\, \Delta\Gamma^{(y)\tau}_{\rho\sigma}, $$

as the inhomogeneous terms cancel.

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$^1$ It is covenient to call $\Gamma^{\lambda}_{\mu\nu}$ Christoffel symbols even if the tangent-space connection $\nabla$ is not torsionfree nor compatible with a metric.